The Bears (2-4) blew a 14-point lead in the second half before coming through in overtime behind Howard, whose 53-yard run put Chicago at the Baltimore 40. After rookie Mitchell Trubisky completed an 18-yard pass to Kendall Wright, Barth delivered the game winner.
Making his first career start on the road, Trubisky directed a conservative game plan that leaned heavily on the run. The first-round draft pick completed 8 of 16 passes for 113 yards and a touchdown.
Howard had 36 carries and the Bears gained 231 yards on 54 attempts.
Baltimore (3-3) trailed 17-3 in the third quarter and 24-16 late in regulation before using two long kick returns to get back in the game.
Bobby Rainey took a kickoff 96 yards for a score to begin the comeback and Michael Campanaro brought back a punt 77 yards for a touchdown with 1:37 remaining. The 2-point conversion was successful, setting up overtime.
The Ravens weren't the only ones to capitalize on big plays. Chicago used a halfback pass from Tarik Cohen to Zach Miller for a first-half score, and Adrian Amosreturned an interception 90 yards for a touchdown for a 24-13 lead with 5:08 left.
Chicago forced three turnovers and frustrated quarterback Joe Flacco throughout the afternoon. Operating without injured receivers Jeremy Maclin (inactive) and Breshad Perriman (second-quarter concussion), Flacco went 24 for 41 for 180 yards with two interceptions.
That put the Ravens in a precarious position, especially with a struggling offense.
Rainey alleviated the pressure by taking the ensuing kickoff for a touchdown. Rainey popped to his feet after being tripped by a teammate, broke right and went the distance.
The Ravens were coming off a 30-17 win in Oakland in which they did not commit a turnover, didn't allow a sack and were penalized only once.
In this one, Baltimore gave the ball away twice, permitted two sacks and was flagged five times - in the first half.
Bears: Host Panthers, who will have a few extra days rest after playing on Thursday night.
Ravens: Face the Vikings on the road for the first time since 2009.
CHICAGO - Case Keenum led three second-half scoring drives in relief of Sam Bradford, and the Minnesota Vikings spoiled prized rookie quarterback Mitchell Trubisky's debut with a 20-17 victory over the Chicago Bears on Monday night.
Kai Forbath kicked a 26-yard field goal in the closing seconds after Harrison Smith intercepted Trubisky, and the Vikings (3-2) came away with the win after dropping two of three.
Called on after Mike Glennon struggled in the first four games, Trubisky played like he belonged. The No. 2 overall pick was 12 of 25 for 128 yards with a touchdown.
The interception came after the Bears (1-4) took over on their 10 with 2:32 left in the game. Smith picked off a pass intended for Zach Miller on the right sideline at the 22.
The Vikings got the ball at the 28, and Forbath booted the winner with 16 seconds left.
Bradford could barely move after missing three straight games because of swelling and soreness in his left knee.
He was just 5 of 11 for 36 yards and was sacked four times - including a safety by Leonard Floyd.
Keenum replaced him with 25 seconds left in the first half and went 17 of 21 for 140 yards and a touchdown.
Jerick McKinnon had a 58-yard touchdown run. Everson Griffen contributed a strip-sack against Trubisky that led to a field goal, and the Vikings beat Chicago on the road for just the third time in 17 games.
The Bears simplified their offense for Trubisky, with fewer receivers and extra blockers at times. But he still showed the mobility and arm strength that convinced general manager Ryan Pace to trade up a spot with San Francisco to grab him on draft night.
Jordan Howard ran for 76 yards. Pat O'Donnell threw a 38-yard touchdown to Benny Cunningham on a fake punt. Floyd had two sacks.
Minnesota was leading 3-2 when Keenum hit Kyle Rudolph for a 13-yard touchdown midway through the third quarter. The Vikings celebrated by playing duck-duck-goose in the end zone.
The Bears lined up to punt on a fourth down at the Minnesota 38 on the next possession. But O'Donnell passed to a wide-open Cunningham, who spun around Marcus Sherels on the way to the end zone, cutting it to 17-9.
Chicago used another trick play to tie it.
The Vikings' Andrew Sendejo tipped a pass right to Zach Miller in the end zone for a 20-yard touchdown. The trickery came on the two-point conversion.
Trubisky handed the ball to Howard, who gave it to Miller. Miller then pitched it to Trubisky, who ran it in on the right side to tie it at 17-all with 12:24 left in the game.